Summary of Mechanical engineering and Engineering mechanics doctoral dissertation: Research on non-linear random vibration by the global – local mean square error criterion

Propose the global – local mean square error criterion (GLOMSEC) for Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system subjected to white noise or color noise excitation. The mean square of response solution will be concentrated to evaluate the accuracy of the proposed criterion by comparison with exact solution or other accepted solutions.
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Summary of Mechanical engineering and Engineering mechanics doctoral dissertation: Research on non-linear random vibration by the global – local mean square error criterion MINISTRY OF EDUCTION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ----------------------------- Nguyen Cao Thang RESEARCH ON NON-LINEAR RANDOM VIBRATION BY THE GLOBAL – LOCAL MEAN SQUARE ERROR CRITERION Major: Engineering Mechanics Code: 9 52 01 01 SUMMARY OF MECHANICAL ENGINEERING AND ENGINEERING MECHANICS DOCTORAL DISSERTATION HANOI - 2019 The dissertation has been completed at: Graduate University of Science and Technology – Vietnam Academy of Science and Technology Supervisors : Dr. Luu Xuan Hung Prof. Dr. Sc. Nguyen Dong Anh Reviewer 1: … Reviewer 2: … Reviewer 3: …. Dissertation is defended at Graduate University of Science and Technology – Vietnam Academy of Science and Technology at …, on date … month … 2019. Hardcopy of the dissertation can be found at: - Library of Graduate University of Science and Technology - Vietnam National Library 1 INTRODUCTION 1. Rationale of the dissertation The analysis, design and control of vibration play an important role in improving effectiveness and performance of structures, vehicles and engines. In recent years, the multi degree of freedom system is used for most of engineering applications. Accordingly, it is necessary to develop Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system, based on dual concept to develop Global local mean square error criterion (GLOMSEC) for MDOF nonlinear system. 2. Object of the dissertation Apply dual concept to solve the limited area [-rx , + rx] in the local mean square error criterion (LOMSEC). By that way, propose the global – local mean square error criterion (GLOMSEC) for Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system subjected to white noise or color noise excitation. The mean square of response solution will be concentrated to evaluate the accuracy of the proposed criterion by comparison with exact solution or other accepted solutions. 3. Research methodology In the dissertation, analyse method, numerical method, Monte – Carlo Simulation method are considered. The analyse method is considered to create the error criterion: base on dual concept in analyse response of nonlinear systems (consider two different approaches to a problem) to obtain the linearized coefficients by close analysis method. The numerical method is considered to program by Matlab software to compute and simulate random nonlinear vibration MDOF systems. The Monte Carlo simulation is considered to find simulation solution for determination the accuracy of linearization method. 4. Scientific and practical application - Develop Gaussian equivalent linearization method (GEL) – one of most popular method used in Random vibration. Particularly, the 2 Global Local Mean Square Error Criterion – GLOMSEC is generalized for MDOF random nonlinear system. - Develop close equation system to determine mean square of responses. Investigate and evaluate the accuracy of proposed criterion for MDOF nonlinear random systems subjected to white noise or color process. - The results of the dissertation are applied to analyse technical nonlinear random systems. 5. Structure of the dissertation The structure of the dissertation includes: the introduction, 4 chapters, the conclusions, a list of publications, the references and the appendix. CHAPTER 1. INTRODUCTION TO PROBABILITY THEORY AND SOME METHODS ANALYSING NON-LINEAR RANDOM VIBRATION 1.1. Random variable and its probabilistic properties Define probability of a random event [29], [69]: Perform n experiments, if the outcome M occurs m times, than probability of outcome M, denote P(M) is the limitation of frequence f(M) = m/n when the number of experiments n increases to infinity: lim f ( M )  P ( M ) (1.1) n  Random variable X is a quantity that links each outcome r of an experiment with a real number X(r) satisfies: a) Set X  x is called an event M for each real number x, b) probability of event X =   equal zero: PX =  = 0 (1.2) The cumulative distribution function (cfd) of the random variable X is defined for any real number x by: 3 F(x) = P[X  x] (1.3) 1.2 Stochastic processes There are definitions of: Probability density function; High order moment; Mathematical expectation; Mean square; Variance; Auto- correlation; covariance. 1.3 Some special stochastic processes There are definitions of: Stationary random processes and Ergodic process; Normal random process or Gaussian process; White noise process; colored noise process; Wiener process and Markov process. 1.4 Some approximately analytical methods for analyzing random oscillation Numerical methods, approximately analytical methods are very popular methods. In detail, there are some useful methods in this dissertation [29-31]: - Perturbation technique. - Fokker-Planck-Kolmogorov (FPK) equation technique. - Stochastic averaging technique. - Statistical linearization technique. 1.5 Fokker-Planck-Kolmogorov (FPK) equation technique and Stochastic averaging technique 1.6 Overview of studies on random oscillations The problem of random vibrations has been studied and presented in many textbooks [26–33]. Oscillation ana ...